An exemption was also given to another amateur, Nick Watney, a senior at Fresno State University.
Two other exemptions will be filled later and will be given to PGA Tour members, officials said.
Barnes is scheduled to play in the Scottish Open July 10-13, and the British Open the following week.
He tied for 59th place at the this year's U.S. Open and generated a following at the Masters when he out-shot Tiger Woods in the first round by seven shots.
Watney has already received an exemption and played in the FBR Capital One Open earlier this month, where he missed the cut in his first pro start.
The RTO at Montreux Golf & Country Club runs Aug. 21-24.
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